English Chess Forum

Note: Copy of report provided to April Council meeting

Back in the autumn, Dave Thomas (the then Manager of Grading and Rating) and myself published a statement on grading anomalies on the ECF website. The conclusions were:
“There is a need for further analysis including impact assessment on grading bands before any action is taken. In particular it is important to give organisers sufficient notice of any such change, and supply them with sufficient data to make reasonable choices about the grading limits they should adopt in events. This will be the responsibility of the grading team and in particular the new Manager of Grading and Rating.”
Since then I took on the role of Manager and Rating, while retaining the role of Director of Home Chess. Wearing two hats has certainly prevented me from giving the grading anomalies the attention that they deserved and other members of the grading team have similarly had other distractions. Consequently, progress has been slower than one would have wished. The plan, which was laid out for the grading team last autumn, contained the follows activities:
1. Conduct statistical analysis of anomaly using chi squared test or equivalent. Establish whether differences between expected and actual performance are within normal statistical variation or whether the deviation can be said to be statistically significant (real data will never provide a perfect fit to a theoretical graph). Note: there are about six years of data in the form of individual games on which analysis can be done.
2. Establish a model to see if the ECF grading methodology is a source of the problem and whether there is a need to change grading methodologies.
3. Establish an appropriate methodology for correcting grades if that is deemed necessary. The methodology needs to include an independent check to determine that the new grades generated are correct.
4. Assess the impact of any necessary changes and see if the impact can be minimised (e.g. so that mid range grades are unaffected).
5. Review how new players and juniors are dealt with in the grading methodology.
6. Propose a set of changes (with the impact on the number of players in each grading band) to ECF Council for approval.

David Welch has made some progress on activity 2 and a technique to confirm the soundness of the current junior increments and new player grade estimation has been established. However, activities 1 3 and 4 remain to be done.

It is now too late for the implementation of significant changes into the 2007 grading list. Consequently, such changes (if necessary) will be implemented in July 2008 (this applies irrespective of whether Council determines to introduce standard play lists from Jan 2008).

I find this grading debate rather interesting.

I read here... The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher. Grades are calculated by averaging out points gained or lost against opponents whose grades are known already. ...and here... Points are scored for each game. For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 40 points, it is taken to be exactly 40 points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade.

According to... The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores.... if player A with grade 'a' and player B with grade 'b' play a match where player A wins with the percentage difference 'd' and where a >= b + d/2, the grade of either player should not change after the match... examples...

A: a=130 d=80-20=60
B: b=100
A's new grade is (20*(100-50) + 80*(100+50)) / 100 = 130 and B's new grade is (20*(130+50) + 80*(130-50)) / 100 = 100, the grades do not change

A: a=150 d=100-0=100
B: b=100
A's new grade is (0*(110-50) + 100*(110+50)) / 100 = 160 and B's new grade is (0*(140+50) + 100*(140-50)) / 100 = 110, the grades do change

A: a=180 d=100-0=100
B: b=100
A's new grade is (0*(140-50) + 100*(140+50)) / 100 = 190 and B's new grade is (0*(140+50) + 100*(140-50)) / 100 = 110, the grades do change

If the rule would read... For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 points, it is taken to be exactly 50 points above (or below) yours.... the grades would not change... examples...

A: a=150 d=100-0=100
B: b=100
A's new grade is (0*(100-50) + 100*(100+50)) / 100 = 150 and B's new grade is (0*(150+50) + 100*(150-50)) / 100 = 100, the grades do not change

A: a=180 d=100-0=100
B: b=100
A's new grade is (0*(130-50) + 100*(130+50)) / 100 = 180 and B's new grade is (0*(150+50) + 100*(150-50)) / 100 = 100, the grades do not change

So, maybe the rule should read... For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 points, it is taken to be exactly 50 points above (or below) yours.... instead of... For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 40 points, it is taken to be exactly 40 points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade.

Please note that in...

A: a=130 d=80-20=60
B: b=100
A's new grade is (20*(100-50) + 80*(100+50)) / 100 = 130 and B's new grade is (20*(130+50) + 80*(130-50)) / 100 = 100, the grades do not change

...we assumed that there were no draws, but say if player B hasn't won...

A: a=130 d=(60+40/2)-(40/2)=60
B: b=100
A's new grade is (40*(100) + 60*(100 + 50)) / 100 = 130 and B's new grade is (40*(130) + 60*(130-50)) / 100 = 100, the grades do not change

...we get the same result.

Apparently, there are also problems in Elo and even Glicko systems. Glicko system has been used on FICS.

Robert, you seem to be over-complicating it. Unless I'm very much mistaken, the reason for the '40 point difference' is to ensure that a stronger player doesn't lose rating points by beating a much weaker player and a weaker player can't gain points by losing to a much stronger player.

They will always get:
(respective to their own grade)

For a win:-
minimum +10
maximum +90

For a loss:-
minimum -10
maximum -90

For a draw:-
their opponent's grade to a maximum of +/- 40

Hello Greg,

Greg Breed wrote:Robert, you seem to be over-complicating it.

I am well known for that.

Greg Breed wrote:the reason for the '40 point difference' is to ensure that a stronger player doesn't lose rating points by beating a much weaker player and a weaker player can't gain points by losing to a much stronger player.

Yes, what I was trying to say is that maybe the rule should define the difference as 50 and not as 40 points, as that would be in accord with The Percentage Rule... The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher.... which I've read about here.

Let me give a simple example... player A with grade 150 plays a 10-game match against player B with grade 100 and wins all the games. According to The Percentage Rule above, the grade of either player should not change after the match (after one calculates new grades for the players including the games from the match), as both had performed as expected from their grades (the difference in percentage scores is 100 and the difference in ratings is 50)... current rules say that for grading purposes the difference in grades should be 40 or less, so A's grade from B's perspective is 110 and B's grade from A's perspective is 140... then you calculate new grades for players A and B...

A: (0*(110-50) + 10*(110+50)) / 10 = 160
B: (0*(140+50) + 10*(140-50)) / 10 = 110,

...and you see that they have changed, what is not in accord with The Percentage Rule, and I thought this might have been the reason for the ECF grade anomalies.

If the rule would define a 50 point difference, i.e. ... For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 (not 40) points, it is taken to be exactly 50 (not 40) points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade....

then A's grade from B's perspective would be 100 and B's grade from A's perspective would 150... and then after you calculate new grades for players A and B:

A: (0*(100-50) + 10*(100+50)) / 10 = 150
B: (0*(150+50) + 10*(150-50)) / 10 = 100,

...you would see that they have not changed, what is in accord with The Percentage Rule.

If we wanted to be precise The Percentage Rule... should have read The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores or higher. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher.... and that is because the rating difference can be greater than 50 which is greater than 100/2 (half the difference in percentage scores) where a 100 is the maximum difference in percentage scores one can achieve (one cannot win more than 100% games)...

In my previous post I stated a general condition in accord with The Percentage Rule where in a match (of as many games) grades of the players should remain unchanged... and that is...

if player A with grade 'a' and player B with grade 'b' play a match which player A wins (or draws) with the percentage difference 'd' and where a >= b + d/2, the grade of either player should not change after the match...

examples of 10-game matches where the grade of the players should not change after the match... according to The Percentage Rule...

1.
A: a=100
B: b=100
A scores 5 points, d=(50-50=0)

2.
A: a=120
B: b=100
A scores 7 points, d=(70-30=40)

3.
A: a=130
B: b=100
A scores 8 points, d=(80-20=60)

4.
A: a=140
B: b=100
A scores 9 points, d=(90-10=80)

5.
A: a=150
B: b=100
A scores 10 points, d=(100-0=100)

6.
A: a=160
B: b=100
A scores 10 points, d=(100-0=100)

7.
A: a=220
B: b=100
A scores 10 points, d=(100-0=100)

...according to the present rules you will see that the grade changes in examples 5 to 7. ... this would not happen if the rule said that the difference was 50 and not 40 points.

If this is true and the cause of grading anomalies, the change in the rules would be trivial.

Of course, I understand that grading/ rating issue is not so simple and that there is a sophisticated criticism (using a complex statistical analysis) which also targets Elo an even Glicko rating systems, but maybe keeping present grading system and making this small change (to correct for the mistakes made when grading games where the difference in grade is more than 40) could be satisfactory.

Hi Robert,

I think I've worked out where you've gone wrong...

Robert Jurjevic wrote:Yes, what I was trying to say is that maybe the rule should define the difference as 50 and not as 40 points, as that would be in accord with The Percentage Rule... The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher.... which I've read about here.

This isn't entirely correct. I think what they mean is that, given a 60% win ratio, you would expect that player to have a grade 30 points higher than his opponent - not the performance rating.

Robert Jurjevic wrote:Let me give a simple example... player A with grade 150 plays a 10-game match against player B with grade 100 and wins all the games. According to The Percentage Rule above, the grade of either player should not change after the match

This is where I think you're being led astray by Wikipedia.

In the above example you've given, player A would gain 10 points to become 160. Because the gap between them is greater than 40 it is taken to be exactly 40, therefore each win is +10, which, when averaged out over 10 games is still +10.

Hello Greg,

Greg Breed wrote:In the above example you've given, player A would gain 10 points to become 160. Because the gap between them is greater than 40 it is taken to be exactly 40, therefore each win is +10, which, when averaged out over 10 games is still +10.

Exactly, that is why I would propose to change the rules as follows...

For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 (not 40) points, it is taken to be exactly 50 (not 40) points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade....

...in order to correct this, what I think is an anomaly.

Do you agree that the grades should have not changed as the players scored (the match percentage scores) as expected according to their grades?

Robert Jurjevic wrote:For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 (not 40) points, it is taken to be exactly 50 (not 40) points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade....

...in order to correct this, what I think is an anomaly.

Do you agree that the grades should have not changed as the players scored (the match percentage scores) as expected according to their grades?

When you put it like that I see what you're saying. I guess what you're saying is: if you're expected to win against someone (+50 difference or more), you shouldn't profit from it.

Have I got it right?

Greg Breed wrote:When you put it like that I see what you're saying. I guess what you're saying is: if you're expected to win against someone, you shouldn't profit from it. Have I got it right?

My understanding is that...

That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher....

in..

The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores or higher. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher....

...means that if player A say with grade a=130 plays a match against player B with grade b=100 (b=130-30) it is expected (according to their grades) that player A will sore 80% in the match giving 80%-20%=60% difference in percentage scores, or vice versa... if player A scored 80% in the match it is expected that his grade is 30 points (i.e. 30=(80%-20%)/2 half the difference in percentage scores) higher (a=130) than the grade of player B (b=100)... and if player A did score 80% (giving 80%-20%=60% difference in percentage scores) in the match the grades of both players should remain unchanged as they performed as expected (according to their grades).

Now three examples follow (calculated using current grading rules)...

1.
A: a=130 d=80-20=60 (A scores 80%)
B: b=100
A's new grade is (20*(100-50) + 80*(100+50)) / 100 = 130 and B's new grade is (20*(130+50) + 80*(130-50)) / 100 = 100, the grades do not change

2.
A: a=150 d=100-0=100 (A scores 100%)
B: b=100
A's new grade is (0*(110-50) + 100*(110+50)) / 100 = 160 and B's new grade is (0*(140+50) + 100*(140-50)) / 100 = 110, the grades do change

3.
A: a=200 d=100-0=100 (A scores 100%)
B: b=100
A's new grade is (0*(160-50) + 100*(160+50)) / 100 = 210 and B's new grade is (0*(160+50) + 100*(160-50)) / 100 = 110, the grades do change

Example 1 is okay, the grades do not change (that is what one would expect from the grading system), but example 2 gives a wrong result (note that in examples 2 an 3 the difference in percentage scores is 100%-0%=100% which gives 50 points of grading difference), as the grades change even though they should not... I think that present system works fine for all games where difference in grades is less than or equal to 40 points, but fails when grade correction is to be applied (40 point difference rule)...

...I showed that if the present grading rules are restated as follows...

For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 (not 40) points, it is taken to be exactly 50 (not 40) points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade....

...the calculation (using a new rule) would give correct results for all, example 1, 2 and 3 (you may tray it)...

Note that I also think that in example 3 the grades should have not changed, as if, according to the grading system, a 150 player is expected to score a 100% against a 100 player then a 200 player is expected even more... true, there is no additional reward for a 100 player for losing all games to 200 player rather than 150 player, but I guess that this is right.

So, if one changes 40 point difference rule into 50 point difference rule one should get a "correct" grading system.

A note on negative grades.

It looks like the present grading system has no upper nor lower grade limit. In that respect a grade of 0 or -50 would be perfectly legal, like say a grade of 500. It only matters what is the difference between the highest grade and the lowest grade (the grade range) and what actual performance one would mark with 0.

One possible way to avoid negative grades could be to increase all grades for some safety margin, say a 100 points, and hope that no grade would ever drop below zero, although this could not be guaranteed regardless of the chosen safety margin.

Please note that 0 has been arbitrarily chosen in other fields than chess ability grading, say temperature scales (0 K is -273.15 C), etc., although temperature is not a good example as there is an absolute zero.

Maybe there are grading systems which may impose upper and lower limits to the grades intrinsically built into the system itself, and I believe in such a system a grade would be in a non-linear relationship with the percentage scores i.e. higher and lower grades would be "compressed".

In my opinion not allowing for zero or negative grades (within the present grading system) may also cause grading anomalies. Let me explain...

Let us assume...

...player A with grade 'a' and player B with grade 'b' play a match which player A loses with the percentage difference 'd', their grades are calculated after the match and player A has a new grade 'a2' and player B 'b2'...

1.
A: a=100 -> a2=0
B: b=50 -> b2=150
A scores 0%, d=(0-100=-100)
B wins all the games and A's new grade is 0 and B's new grade is 150 (the grades are calculated with ammended rules where the difference is defined as 50 and not 40 points, please see my previous post for the discussion on that rule change)

2.
A: a=50 -> a2=-50
B: b=0 -> b2=100
A scores 0%, d=(0-100=-100)
B wins all the games and A's new grade is -50 and B's new grade is 100 (the grades are calculated with ammended rules where the difference is defined as 50 and not 40 points, please see my previous post for the discussion on that rule change)

... if in the example 2 you forced (against the calculation) A's grade to be a2=0 (rather than a2=-50) that would be the same as you would have forced (against the calculation) A's grade in example 1 to be a2=50 (rather than a2=0), unsoundly increasing A's greade for 50, and therefore in my opinion not allowing negative grades should inflict on grading system.

To summarize (from my side)... in order to "fix" the grading system on should (in my opinion)...

• change the grading rules to read... For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 (not 40) points, it is taken to be exactly 50 (not 40) points above (or below) yours.
• allow for zero and negative grades (all grades may be artificially increased for the same amount in hope that after the change no grade will ever fall to zero or below, but cutoffs on the lower end should not be made, even if they fall below zero)

The problem with negative grades (which usually occur with new juniors) is that it's demoralising for them. The grades last for a whole year and it can put them off the game.

A minor point maybe, but I bet you i'm not far off the mark.

Other than that, you do have a point, the rest of it is beyond my simple brain unfortunately. If I keep it simple I can work it out. I'll leave the machinations to those further up the hierachy than me

I think that I may have found a big problem in grading mechanism...

Let player A with grade 130 scores 80% (say 8 wins and 2 loses in 10 games) in a match against player B with grade 100. According to the grading theory the players performed as expected (130-100 = (2*80%-100)/2) and the grade of either player should not change after the match, no problem here, as the new grades would be exactly the same as the old ones (you can check that by calculation).

If payer A had scored 70% (say 7 wins and 3 loses in 10 games) in the match (rather than 80%) then the new grades would be 120 and 110 (you can check that by calculation as well). I think that the grades should have been 125 and 105 (rather than 120 and 110). The reason for this assumption is that if they had played a second match (right after the first one) and if player A scored 70% again, according to the grading theory, the grades after the second match would have remained unchanged (you can check that by calculation if you assume that the grades after the fist match were 125 and 105), while if the grades were 120 and 110 they would change (after the second match) to 130 and 100 back. This does not make sense, as player A had surely under-performed in both of the matches while still his grade or his opponent's remained unchanged (after two matches of 70% performance for player A). Also, if you have graded both matches in one go the grades would be 120 and 110.

Let us call this problem for future reference a game subset grading problem (as grading of two matches separately will give a different result than grading both matches in one go).

Do you agree with this (or I miscalculated or misinterpreted something)?

I will try to find a solution to this problem (maybe there isn't one, i.e. you can find the fix for a set of individual cases but not one in general), so that the calculated grades are 125 and 105 rather than 120 and 110, still satisfying the rule... The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher. Grades are calculated by averaging out points gained or lost against opponents whose grades are known already.

Maybe this is not a real problem as grading is done on a relatively big chunk of games (i.e. like you have graded both matches in one go), but this discussion may suggest that if grades are calculated twice or more times a year we might see some effects of this anomaly.

Introduction...

I have found a new grading system which resolves the game subset grading problem (mentioned in one of my previous posts). Now I'm going to elaborate all, step by step (be warned this is a long post, believe me I tried to keep it as short as possible, ha, maybe my saying should be: Things should be made as short as possible, but not any shorter to paraphrase a famous Einstein's quotation Things should be made as simple as possible, but not any simpler.)...

Conditions...

Condition 0: The grades should not be compressed either on low or high end of the grade range (the compression can be imposed, for example, if one does not allow for zero an negative grades).

Condition 1: The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores or higher. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher.

Condition 1 can be restated as follows:

Condition 1: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). If the players play a match it is expected (according to their grades) that player A performance 'p' (in percent) will be p = 50+a-b, or equivalently if performance of player A is 'p', p = 50+a-b, the players performed as expected (according to their grades) and their grades would remain unchanged after the match. For example, if a=130 and b=100 and A performs 80% in the match the grades will not change after the match (A's expected performance is p = 50+a-b = 50+130-100 = 80%).

Condition 2: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). If the players play a match and if performance of player A is 'p2', not necessarily equal to the expected performance, p2 /= p = 50+a-b, the grades calculated after the match, 'a2' and 'b2', must be such that if the players had played a rematch and performance of player A was again 'p2', the grades after the rematch, 'a3' and 'b3', should be the same as the grades after the match, a3=a2 and b3=b2. For example, if a=130 and b=100 and A performs 70% in both, the match and rematch, the grades after the match, 'a2' and 'b2', and rematch, 'a3' and 'b3', should be the same, say Amended Grading System (defined below) would give, a2=125 and b2=105, and, a3=125 and b3=105, note that present grading system as well as Corrected Grading System (defined below) would give a2=120 and b2=110, and, a3=130 and b3=100.

Rules...

Rule 0: There should not be imposed any upper or lower limit on the grades (negative and zero grades should be allowed for).

Rule 1a: For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 40 points, it is taken to be exactly not 40 points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade.

Rule 1b: For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 (not 40) points, it is taken to be exactly 50 (not 40) points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade.

Rule 2: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). Calculate grade difference 'd', d = a-b >= 0. Let g = 50 ('g' is a constant). Calculate base grades 'a0' and 'b0', if d > g then a0 = b+g and b0 = a-g else a0 = a and b0 = b. Calculate base grade difference 'd0', if d > g then d0 = g else d0 = d. Calculate grade gains 'gw', 'gd' and 'gl', gw = (50+d0)/2 and gd = d0/2 and gl = (50-d0)/2, Calculate new grades 'a2' and 'b2', if A wins (B loses) a2 = b0+gw and b2 = a0-gw, if it is a draw a2 = b0+gd and b2 = a0-gd, if A loses (B wins) a2 = b0-gl and b2 = a0+gl.

In order to be able to compare Rule 2 with Rules 1a and 1b, let us express Rules 1a and 1b in algebraic form:

Rule 1a: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). Calculate grade difference 'd', d = a-b >= 0. Let g = 40 ('g' is a constant). Calculate base grades 'a0' and 'b0', if d > g then a0 = b+g and b0 = a-g else a0 = a and b0 = b, Calculate new grades 'a2' and 'b2', if A wins (B loses) a2 = b0+50 and b2 = a0-50, if it is a draw a2 = b0 and b2 = a0, if A loses (B wins) a2 = b0-50 and b2 = a0+50.

Rule 1b: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). Calculate grade difference 'd', d = a-b >= 0. Let g = 50 ('g' is a constant). Calculate base grades 'a0' and 'b0', if d > g then a0 = b+g and b0 = a-g else a0 = a and b0 = b, Calculate new grades 'a2' and 'b2', if A wins (B loses) a2 = b0+50 and b2 = a0-50, if it is a draw a2 = b0 and b2 = a0, if A loses (B wins) a2 = b0-50 and b2 = a0+50.

We can see that (only) additional complexity of Rule 2 (in comparison to Rules 1a and 1b) is the calculation of base grade difference and grade gains, and using them in the calculation of the new grades.

Grading systems...

Let us define three grading systems:

1. Grading System This is a current grading system. The grades are calculated using Rule 1a. Rule 0 does not apply. The system does not satisfy any of the above mentioned Conditions (Condition 1 is satisfied in all cases where the grade difference is less than or equal to 40).

2. Corrected Grading System This is a suggested simple modification of the current grading system in order to fix the flaws in the Grading System. The grades are calculated using Rule 1b. Rule 0 applies. The system satisfies Conditions 0 and 1.

3. Amended Grading System This is a suggested modification of the current grading system in order to make further improvement of the Corrected Grading System (Amended Grading System does not suffer from game subset grading problem, i.e., it does satisfy Condition 2). The grades are calculated using Rule 2. Rule 0 applies. The system satisfies Conditions 0, 1 and 2.

Formulas...

Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). Let the players play a match in which player A scores 'nw' wins, 'nd' draws and 'nl' loses, then the formulas for calculating the grades, 'a2' and 'b2', after the match are (these can be copied directly into Mathematica computer program or calculated by hand, input parameters are: 'a', 'b', 'nw', 'nd', 'nl'):
Questions and answers...

Q: Why is the Corrected Grading System better than the Grading System?
A: Because the Grading System even does not satisfy the basic condition of the ECF grading system which is expressed in Condition 1. The Grading System also compresses the grades by not allowing for negative grades.

Q: Why is the Amended Grading System better than the Corrected Grading System?
A: Because the Corrected Grading System suffers from game subset grading problem (it does not satisfy Condition 2) and the Amended Grading System does not (it satisfies Condition 2).

Q: Is the Amended Grading System correct (i.e., if one uses Rules 0 and 2 for grade calculation will all Conditions, 0, 1 and 2, be satisfied)?
A: Most likely yes (Rule 2 is derived from general mathematical principles) but it has to be checked in practice (people are invited to try to find an example which will falsify the theory).

Q: Is the game subset grading problem really a problem?
A: It may not be, but the following example shows that the system which does not suffer from it might be better: Player A and B with grades a=130 and b=100 play a match and then afterwards a rematch, in both, the match and the rematch, player A scores 70%, the calculated grades after the match, 'a2' and 'b2', and rematch, 'a3' and 'b3', are as follows, the Grading System and the Corrected Grading System give, a2=120 and b2=110, and, a3=130 and b3=100, the Amended Grading System gives, a2=125 and b2=105, and, a3=125 and b3=105, the calculated grades after the match and rematch taken as one match, 'a4' and 'b4', are as follows, the Grading System and the Corrected Grading System give, a4=120 and b4=110, the Amended Grading System gives, a4=125 and b4=105, looks plausible that the grading system should give, a4=a3 and b4=b3.

Q: How much is the Amended Grading System more complex than the Grading System or the Corrected Grading System?
A: This is the same as asking how much is Rule 2 more complex than Rules 1a or 1b, and the answer is that additional complexity in Rule 2 (in comparison to Rules 1a or 1b) is the calculation of base grade difference and grade gains, and using them in the calculation of the new grades.

Falsification...

Although the Amended Grading System's Rule 2 is derived from general mathematical principles (say, I mathematically proved that two draws should count as one win in all circumstances, that Condition 2 should be satisfied for all cases in general, etc.) you are welcome to challenge the system by finding falsification examples, i.e., to try to find an example which will falsify the theory (those of you who are interested in philosophy and know the work of Karl Popper word recognize that this is, according to Popper, exactly how one should prove the correctness of a new scientific theory, as according to Popper a theory which can never be falsified is not a scientific theory, i.e., astrology for example is not a scientific theory according to Popper, of course, if correct, the Amended Grading System should survive all of the initial attacks and has to obey all Conditions, 0, 1 and 2 in general, but it may be falsified if someone finds, say, Condition 3, which ought to be obeyed but it is not, then a new system which would be an extension of the Amended Grading System can be advised).

Conclusions...

In my opinion, it might be worth trying to test, and if proved to be correct (corrections might be possible if flaws are found), to implement the Amended Grading System. If, on the other hand, the Amended Grading System looks too complex to be used in practice, then, in my opinion, the implementation of the Corrected Grading System is a must (though Rule 0 may be chosen not to apply if so desired).

What follows....

Let us call Grading System, GS, Corrected Grading System CGS and Amended Grading System AGS. I already have mathematical proofs that AGS satisfies Condition 1 (in general) and that two draws count as one win in general (i.e. the grade after a match of two players depend only on the match percentage score and no on the number of wins, draws and loses, this holds for any conceivable match and combination of the number of wins, draws and loses which give the same match percentage score). Mathematical proof that AGS satisfies Condition 2 is yet to be found. I have found a very simple form of Rule 2, it will be presented. I will present some calculation examples for AGS and compare the results with CGS and GS.

Heavens above, this is so boring!

A simple form of Rule 2....

I found that Rule 2...

Rule 2: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). Calculate grade difference 'd', d = a-b >= 0. Let g = 50 ('g' is a constant). Calculate base grades 'a0' and 'b0', if d > g then a0 = b+g and b0 = a-g else a0 = a and b0 = b, Calculate grade gains 'gw', 'gd' and 'gl', if d > g then gw = g and gd = 0 and gl = g and else gw = (50+d)/2 and gd = d/2 and gl = (50-d)/2, Calculate new grades 'a2' and 'b2', if A wins (B loses) a2 = b0+gw and b2 = a0-gw, if it is a draw a2 = b0+gd and b2 = a0-gd, if A loses (B wins) a2 = b0-gl and b2 = a0+gl.

...can be expressed in simple form as...

Rule 2: For a win you score average grade plus 25; for a draw, average grade; and for a loss, average grade minus 25. Note that, if your opponent's grade differs from yours by more than 50 points, it is taken to be exactly 50 points above (or below) yours. Average grade is half of the sum of your and your opponent's grade. At the end of the season an average of points-per-game is taken, and that is your new grade.

This shows that calculation of grades with AGS, which uses Rule 2, would be equally easy as with GS or CGS...

GS uses Rule 1a...

Rule 1a: For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 40 points, it is taken to be exactly not 40 points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade.

CGS uses Rule 1b...

Rule 1b: For a win you score your opponent's grade plus 50; for a draw, your opponent's grade; and for a loss, your opponent's grade minus 50. Note that, if your opponent's grade differs from yours by more than 50 (not 40) points, it is taken to be exactly 50 (not 40) points above (or below) yours. At the end of the season an average of points-per-game is taken, and that is your new grade.

Examples of grade calculation....

Let us assume that your grade is a 100 and that you play opponent of...

1a. ...grade 20 and win... your new grade according to... GS is 60+50 = 110... CGS is 50+50 = 100... and AGS is (100+50)/2+25 = 100...
1b. ...grade 20 and draw... your new grade according to... GS is 60... CGS is 50... and AGS is (100+50)/2 = 75...
1c. ...grade 20 and lose... your new grade according to... GS is 60-50 = 10... CGS is 50-50 = 0... and AGS is (100+50)/2-25 = 50...

2a. ...grade 50 and win... your new grade according to... GS is 60+50 = 110... CGS is 50+50 = 100... and AGS is (100+50)/2+25 = 100...
2b. ...grade 50 and draw... your new grade according to... GS is 60... CGS is 50... and AGS is (100+50)/2 = 75...
2c. ...grade 50 and lose... your new grade according to... GS is 60-50 = 10... CGS is 50-50 = 0... and AGS is (100+50)/2-25 = 50...

3a. ...grade 60 and win... your new grade according to... GS is 60+50 = 110... CGS is 60+50 = 110... and AGS is (100+60)/2+25 = 105...
3b. ...grade 60 and draw... your new grade according to... GS is 60... CGS is 60... and AGS is (100+60)/2 = 80...
3c. ...grade 60 and lose... your new grade according to... GS is 60-50 = 10... CGS is 60-50 = 10... and AGS is (100+60)/2-25 = 55...

4a. ...grade 100 and win... your new grade according to... GS is 100+50 = 150... CGS is 100+50 = 150... and AGS is (100+100)/2+25 = 125...
4b. ...grade 100 and draw... your new grade according to... GS is 100... CGS is 100... and AGS is (100+100)/2 = 100...
4c. ...grade 100 and lose... your new grade according to... GS is 100-50 = 50... CGS is 100-50 = 50... and AGS is (100+100)/2-25 = 75...

5a. ...grade 140 and win... your new grade according to... GS is 140+50 = 190... CGS is 140+50 = 190... and AGS is (100+140)/2+25 = 145...
5b. ...grade 140 and draw... your new grade according to... GS is 140... CGS is 140... and AGS is (100+140)/2 = 120...
5c. ...grade 140 and lose... your new grade according to... GS is 140-50 = 90... CGS is 140-50 = 90... and AGS is (100+140)/2-25 = 95...

6a. ...grade 150 and win... your new grade according to... GS is 140+50 = 190... CGS is 150+50 = 200... and AGS is (100+150)/2+25 = 150...
6b. ...grade 150 and draw... your new grade according to... GS is 140... CGS is 150... and AGS is (100+150)/2 = 125...
6c. ...grade 150 and lose... your new grade according to... GS is 140-50 = 90... CGS is 150-50 = 100... and AGS is (100+150)/2-25 = 100...

7a. ...grade 180 and win... your new grade according to... GS is 140+50 = 190... CGS is 150+50 = 200... and AGS is (100+150)/2+25 = 150...
7b. ...grade 180 and draw... your new grade according to... GS is 140... CGS is 150... and AGS is (100+150)/2 = 125...
7c. ...grade 180 and lose... your new grade according to... GS is 140-50 = 90... CGS is 150-50 = 100... and AGS is (100+150)/2-25 = 100...

Mathematical details...

Condition 1: The basis of English (ECF) ratings broadly speaking is that the difference in ratings is half the difference in percentage scores or higher. That is, if player A beats player B in a match 8 to 2 (60% difference), you would expect his grade to be about 30 points higher.

...or...

Condition 1: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). If the players play a match it is expected (according to their grades) that performance of player A 'p' (in percent) will be p = 50+a-b, or equivalently if performance of player A is 'p', p = 50+a-b, the players performed as expected (according to their grades) and their grades would remain unchanged after the match. For example, if a=130 and b=100 and A performs 80% in the match the grades will not change after the match (A's expected performance is p = 50+a-b = 50+130-100 = 80%).

Condition 2: Let 'a' and 'b' are grades of two players, A and B, and let a >= b ('a' must be greater than or equal to 'b'). If the players play a match and then afterwards a rematch and if performance of player A in both, the match and rematch, is 'p2', not necessarily equal to the expected performance, p2 /= p = 50+a-b ('/=' means 'not equal'), if the players were graded (twice) after each match, and (once) after the match and rematch taken as one match, one should obtain the same final grades (for both players). For example, player A and B with grades a=130 and b=100 play a match and then afterwards a rematch, in both, the match and the rematch, player A scores 70%, the calculated grades after the match, 'a2' and 'b2', and rematch (we use 'a2' and 'b2' in calculation), 'a3' and 'b3', are as follows, GS and CGS give, a2=120 and b2=110, and, a3=130 and b3=100, AGS gives, a2=125 and b2=105, and, a3=125 and b3=105, the calculated grades after the match and rematch taken as one match, 'a4' and 'b4', are as follows, GS and CGS give, a4=120 and b4=110, AGS gives, a4=125 and b4=105, Condition 2 requires that the grading system gives a4=a3 and b4=b3.

Condition 1 is a basic principle of ECF grading system and has to be obeyed.

Condition 2 is my extension.

I have mathematically proven that AGS counts two draws the same as one win (i.e. the grade after a match of two players depend only on the match percentage score and not on the number of wins, draws and loses, this holds for any conceivable match and combination of the number of wins, draws and loses which give the same match percentage score)... (if anybody wants details on the proof please let me know)... (Maybe it is necessary to prove that this holds not only in the match of two players but in a more general case, say for a player on a tournament, etc.)

I have mathematically proven that AGS obeys Condition 1... (if anybody wants details on the proof please let me know)... What is left to be mathematically proven is that AGS obeys Condition 2 (and perhaps a more general form of treating two draws the same as one win).

Why AGS is the best (of the there)...

AGS obeys Condition 1 and most likely Condition 2 (it remains to mathematically prove that AGS obeys Condition 2 in general, but examples suggests that it does). Neither GS nor CGS obey Condition 2 (there are examples which can prove that).

CGS obeys Condition 1 but it does not obey Condition 2.

GS does not obey Condition 1 nor Condition 2 (it obeys Condition 1 in all cases where the difference in grades is 40 points or less).

Mathematicians...
Are there mathematicians who are willing to check my proofs or to do some missing proofs? The idea is to prove that AGS is correct (and better than GS and CGS) and adopt it as a new grading system of ECF. Thanks.

AGS...
What do you think about AGS (Amended Grading Sytem)? Thanks.

By definition AGS uses...

Rule 2: For a win you score average grade plus 25; for a draw, average grade; and for a loss, average grade minus 25. Note that, if your opponent's grade differs from yours by more than 50 points, it is taken to be exactly 50 points above (or below) yours. Average grade is half of the sum of your and your opponent's grade. At the end of the season an average of points-per-game is taken, and that is your new grade.